Optimal transport on spherical manifolds has been extensively studied from theoretical standpoints. [1]}, [2]}, [3]} study the regularity (continuity, smoothness) of transport maps on spheres and other non-negatively curved manifolds. Regularity and smoothness are more intricate on negatively curved manifolds, e.g. hyperbolic spaces. Nevertheless, several works demonstrated that transport can be made smooth through a minor change to the Riemannian cost [4]}.
[5]}, [6]} leverage this to learn transport
maps on hyperbolic spaces, in which case maps are parameterized as
hyperbolic neural networks.
